Projection and rescaling algorithm for finding maximum support solutions to polyhedral conic systems

Abstract

We propose a simple projection and rescaling algorithm that finds maximum support solutions to the pair of feasibility problems \[ \text{find} \; x\in L\cap\mathbb{R}^n_{+} \;\;\;\; \text{ and } \; \;\;\;\; \text{find} \; \hat x\in L^\perp\cap\mathbb{R}^n_{+}, \] where $L$ is a linear subspace of $\mathbb{R}^n$ and $L^\perp$ is its orthogonal complement. The algorithm complements a basic procedure that involves only projections onto $L$ and $L^\perp$ with a periodic rescaling step. The number of rescaling steps and thus overall computational work performed by the algorithm are bounded above in terms of a condition measure of the above pair of problems. Our algorithm is a natural but significant extension of a previous projection and rescaling algorithm that finds a solution to the problem \[ \text{find} \; x\in L\cap\mathbb{R}^n_{++} \] when this problem is feasible. As a byproduct of our new developments, we obtain a sharper analysis of the projection and rescaling algorithm in the latter special case.